AI 中文总结
研究缠绕 2 - 群\(H\)与循环群\(K\)的扎帕 - 塞普积\(G = HK\)，当\(H\)的基\(B\)在\(G\)中正规时，分情况通过多项式同余式对其进行分类，包括对称循环矩阵情况及一般情况，并单独处理模为 4 的分类。
AI 中文摘要
设\(n,m\geq1\)，令\(H = C_{2^n}\wr C_2\)为缠绕 2 - 群，\(K = C_{2^m}=\langle z \rangle\)为循环群。我们对扎帕 - 塞普积\(G = HK\)进行分类，其中\(H\)的基\(B\cong C_{2^n}\times C_{2^n}\)在\(G\)中正规。当\(z\)对\(B\)作用的矩阵\(Z\)与两个基生成元的交换\(J\)可交换（即\(Z\)是对称循环矩阵）时，我们通过一个关于元组\((p,q,r,s,c)\)的七个多项式同余式的显式系统对这些积进行分类。去掉这个假设，我们通过关于\(Z\)的元素和参数\((r,s,c)\)的五个同余式对所有\(B\)正规的此类积进行统一分类，对称情形是特殊情况\(JZ = ZJ\)。最后，我们单独处理模\(M = 2^m = 4\)的分类，因为在此情况下同余式退化。
英文摘要
Let \(n, m \ge 1\), and let \(H = C_{2^n}\wr C_2\) be the wreathed \(2\)-group and \(K = C_{2^m}=\langle z \rangle\) a cyclic group. We classify the Zappa--Szép products \(G = HK\) in which the base \(B \cong C_{2^n}\times C_{2^n}\) of \(H\) is normal in \(G\). When the matrix \(Z\) of the \(z\)-action on \(B\) commutes with the swap \(J\) of the two base generators -- equivalently, \(Z\) is symmetric circulant -- we classify these products by an explicit system of seven polynomial congruences in a tuple \((p,q,r,s,c)\). Dropping this hypothesis, we obtain a unified classification of all such products with \(B\) normal by five congruences on the entries of \(Z\) and the parameters \((r,s,c)\), of which the symmetric case is the specialisation \(JZ = ZJ\). Finally, we separately treat the classification for the modulus \(M = 2^m = 4\), since in this case the congruences degenerate.
